Permutation & Combination

Formula

nPr = n!/(n-r)! | nCr = n!/(r!(n-r)!)

nPr calculates arrangements where order matters. nCr calculates selections where order doesn't matter.

Worked Examples

Example 1: Race Winners

Problem:8 runners, top 3 places

Solution:Order matters (Gold, Silver, Bronze). 8P3 = 8!/5! = 8×7×6 = 336.

Result:336 ways

Example 2: Card Hand

Problem:52 cards, deal 5

Solution:Order doesn't matter. 52C5 = 2,598,960.

Result:2,598,960 hands

Frequently Asked Questions

What is the difference between Permutation and Combination?

Order matters in Permutations (arrangements like a lock code). Order does NOT matter in Combinations (groups like a fruit salad).

Which result is always larger?

nPr (Permutations) is always larger than or equal to nCr because each combination can be arranged in multiple orders.

What is a Circular Permutation?

Arranging items in a circle. Since there is no "start," the formula is (n-1)!. Permutation & Combination finds linear permutations.

What is Pascal's Triangle?

A geometric arrangement of numbers where each number is the sum of the two above it. The entries correspond to Combination values (nCr).

How do combinations and permutations differ?

Combinations count selections where order does not matter — choosing 5 cards from 52 is a combination (C(52,5) = 2,598,960). Permutations count arrangements where order matters — arranging 5 cards in a specific sequence is a permutation. Lottery jackpots use combinations; ranking finishers in a race uses permutations.

Background & Theory

Permutations (Order Matters):
Arranging items in a specific sequence.

Formula: nPr = n! / (n - r)!
Example: Lock codes, race winners (1st, 2nd, 3rd).

Combinations (Order Doesn't Matter):
Selecting items where the sequence is irrelevant.

Formula: nCr = n! / (r! (n - r)!)
Example: Lottery numbers, picking a team of players.

Factorial (!):
n! = n × (n-1) × ... × 1. (0! = 1)

History

Combinatorics, the mathematics of counting, dates back to ancient India and China. The Jain mathematician Mahavira (c. 850 AD) gave formulas for permutations and combinations. In Europe, the Jewish mathematician Levi ben Gerson (1321) discovered the formula for permutations. Blaise Pascal (17th century) popularized the "Pascal's Triangle," which contains the values for combinations (nCr).